A survey of complex generalized weighing matrices and a construction of quantum error-correcting codes

TL;DR

This paper surveys complex generalized weighing matrices, explores their existence conditions, and demonstrates their application in constructing quantum error-correcting codes, highlighting their potential in quantum computing and related fields.

Contribution

It provides a comprehensive overview of complex generalized weighing matrices, extends existence conditions, and constructs new quantum codes demonstrating their practical utility.

Findings

Extended known existence conditions for these matrices

Compiled existence results for small parameters

Constructed interesting quantum error-correcting codes

Abstract

Some combinatorial designs, such as Hadamard matrices, have been extensively researched and are familiar to readers across the spectrum of Science and Engineering. They arise in diverse fields such as cryptography, communication theory, and quantum computing. Objects like this also lend themselves to compelling mathematics problems, such as the Hadamard conjecture. However, complex generalized weighing matrices, which generalize Hadamard matrices, have not received anything like the same level of scrutiny. Motivated by an application to the construction of quantum error-correcting codes, which we outline in the latter sections of this paper, we survey the existing literature on complex generalized weighing matrices. We discuss and extend upon the known existence conditions and constructions, and compile known existence results for small parameters. Some interesting quantum codes are constructed to demonstrate their value.